11/22/2021

Meet … Enhydra lutris

sea otter

  • Cute! (but are they really!?)
  • Furry! (guess how may hairs per in2? … (hint, humans are born with ~100,000 total)
  • 150,000 hairs / cm2 (>1,000,000 / in2)

Sea otters: keystone species

sea otter keystone

(Estes et al. 1974)

Sea otters: Keystone

sea otter keystone

(Smith et al. 2021)

Sea otters: Range

sea otter range

  • Entire (littorally) North Pacific

Sea otters: Furriness > Cuteness

fur trade

  • Fur trade (Russian -> British -> American) leads to near extirpation across the entire range.
  • 300,000 in 1740 … < 2,000 in 1900.

  • Displacement and indenturing of Indigeneous fishermen (esp. Aleut)

fur trade

… the rush for the otters’ “soft gold” was a predictable boom and bust cycle, a cautionary example of unsustainable resource use, and a socioeconomic driver of Western—mainly American—involvement in the Pacific region starting in the eighteenth century. (Loshbaugh 2021)

Sea otters: curiously not totemic

fur trade art by John Livingston

  • Ainu - Esaman
  • Aleut - Chngatux
  • Alutiiq - Arhnaq
  • Tlingit - Yáxwchʼ
  • Haida - Ku
  • Nuu chah nulth - Кwak̕aƛ
  • Siletz - Elakha

Sea otters: But culturally significant

Sea otter reintroduction: Pacific NW

Remnant populations from Aleutian Islands … released in OR, WA, BC and SE-AK 1969 – 1972.

Reintroduction trade Reintroduction trade

Sea otter reintroduction: Washington State …

Successful!

Population ecology is all about …

\(\huge N\)
  • but where? when?

Here! Now! …

\(\huge N_t\)
  • but how many were there?

That many, then (\(\Delta t\) ago)!

\[\Large N_t = N_{t - \Delta t} + \Delta N\]

slight rearrangement:

\[\Large N_{t+1} = N_t + \Delta N\] For now, we just \(\Delta t = 1\), i.e. it’s the discrete unit that we measure population change. VERY TYPICALLY - whether because of biology or field seasons: \[\Delta t = 1\,\, \textrm{year}\].

How does population change?

\(\Large N_{t+1} = N_t + B - D + I - E\)

Birth

Death

Immigration

Emigration

Assumption 1: No one’s getting on or off the bus

\(\Large N_{t+1} = N_t + B - D\)

Birth

Death

Immigration

Emigration

This is a closed population … and what we will be (mainly) dealing with for the next 3 weeks.

Assumption 2: the important one

The number of Births and Deaths is proportional to N.

\[\Large N_{t+1} = N_t + bN_t - dN_t\] What does that mean?

  • Every female gives birth to the same number of offspring?
  • Every female has the same probability of giving birth?
  • Every female has the same probability of giving birth to the same distribution of offspring?
  • A fixed proportion of all individuals dies?
  • Every individual has the same probability of dying?
  • the distribution of probabilities of dying is constant?

Some math ….

Redefine \(\lambda = b - d:\)

\[N_{t+1} = N_t + r_0 N_t\] \[N_{t+1} = (1 + r_0) N_t\]

\(r_0\) intrinsic growth, i.e. proportion increase per unit time).

\[N_{t+1} = \lambda N_t\]

Cranking this forward

\[N_{t+1} = \lambda(N_t)\] \[N_{t+2} = \lambda(N_{t+1}) = \lambda^2 N_t\] \[N_{t+3} = \lambda^3 N_t\]

\[N_{t+y} = \lambda^y N_t\]
Geometric (same as Exponential) growth.

Exponential growth can be very very very fast

Discrete Model to Continuous Model

Let’s do some trickery, starting with: \[N_{t+1} = (1 + r_0) N_t\] \[N_{t+1} - N_t = r_0 N_t\] \[N_{t+\Delta t} - N_t = r_{\Delta t} N_t\]

\[\lim_{\Delta t \to 0} {\Delta N \over \Delta t} = \lim_{\Delta t \to 0} {r_{\Delta t} \over \Delta t} N\]

Magically define: \({r_\Delta \over \Delta t} = r\) and rewrite \(\Delta\) as \(d\):

\[\large {dN \over dt} = r N\]

Solving this model ….

Start: \[{dN \over dt} = r N\]

Calculate

\(\begin{align} {1\over N} dN &= r dt \\ \int_{t' = t_0}^t {1 \over N(t)} dN &= \int_{t' = t_0}^t r dt \\ \log(N) &= rt + C_0 \\ N &= e^{rt + C_0} \\ \end{align}\)

Discrete vs. Continuous Modeling

Difference equations:

\[N_{t+\Delta t} - N_t = \lambda_{\Delta t} N_t\]

think of absolute change

Pros:

  • Reflects (often) biological reproduction patterns, practical sampling schedule (esp. annual)
  • Intuitive

Cons:

  • Depends on discretization timescale
  • Analytically surprisingly difficult to analyze

Differential equations

\[{dN \over dt} = r N\]

think of rates (change/time).

Pros:

  • Easier “elegant” mathematical analysis
  • Scales nicely

Cons:

  • Unbiological
  • Unintuitive

Sea otter data:

Plot Data:

Fit a line

Plot on Log scale:

Fit linear model log-growth

## 
## Call:
## lm(formula = log(count) ~ I(year - 1970), data = WA)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.191084 -0.062944 -0.005104  0.055518  0.231704 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    4.082641   0.073024   55.91   <2e-16 ***
## I(year - 1970) 0.073251   0.002367   30.95   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1094 on 23 degrees of freedom
## Multiple R-squared:  0.9766, Adjusted R-squared:  0.9755 
## F-statistic: 958.1 on 1 and 23 DF,  p-value: < 2.2e-16

A little math:

\[\log(N_i) = \alpha + \beta \, Y_i\] \[N_i = \exp(\alpha) \times \exp(\beta \, Y_i)\] \[N_i = e^\alpha {e^\beta}^{Y_i}\] \[N_i = N_0 \lambda ^ {Y_i}\]

where \(N_0 = e^{\alpha} = e^{4.08} = 59.14\), and \(\lambda = e^{\beta} = e^{0.07325} = 1.076\).

SO … percent rate of growth is about 7.6%!

Plot exponential growth

References

  • J. A. Estes, J. F. Palmisano. 1974. Sea otters: Their role in structuring nearshore communities. Science 185, 1058–1060.
  • Smith et al. 2021. Behavioral responses across a mosaic of ecosystem states restructure a sea otter–urchin trophic cascade. PNAS Mar 2021, 118 (11)
  • Loshbaugh S. 2021. Sea Otters and the Maritime Fur Trade. In: Davis R.W., Pagano A.M. (eds) Ethology and Behavioral Ecology of Sea Otters and Polar Bears. Ethology and Behavioral Ecology of Marine Mammals.
  • Gilkinson, A.K., Pearson, H.C., Weltz, F. and Davis, R.W., 2007. Photo‐identification of sea otters using nose scars. The Journal of Wildlife Management, 71(6), pp.2045-2051.